Imitation Dynamics in Oligopoly Games with Heterogeneous Players

Transcription

1 THEMA Working Paper n Université de Cergy-Pontoise, France Imitation Dynamics in Oligopoly Games with Heterogeneous Players Daan Lindeman, Marius I. Ochea December 208

2 Imitation Dynamics in Oligopoly Games with Heterogeneous Players Daan Lindeman Marius I. Ochea y th December 208 Abstract We investigate the role and performance of imitative behaviour in a class of quantitysetting Cournot games. Within a framework of evolutionary competition between rational, best-response and imitators players we found that the equilibrium stability depends on the intensity of the evolutionary pressure and on the stability of the cheapest heuristic. When the cheapest behavioural rule is the stable heuristic (i.e. imitation), the dynamics converge to a situation where most rms use this behavioural rule and all rms produce the Cournot-Nash equilibrium quantity. When the cheapest heuristic is unstable one (i.e. best-response), complicated endogenous uctuations may occur along with the co-eistence of heuristics. JEL classi cation: C72, C73, D43 Keywords: Competing heuristics, Imitation, Evolutionary dynamics Loge, Amsterdam: daan.lindeman@loge.nl. y THEMA, Université de Cergy-Pontoise, 33, bd. marius.ochea@u-cergy.fr du Port, 950, Cergy-Pontoise Cede, France.

3 Introduction Theocharis (960) remarkably shows that, when rms compete on quantity using the Cournot (838) adjustment process, the Cournot-Nash equilibrium becomes unstable if the number of rms eceeds two. In fact, with linear demand and constant marginal costs, the Cournot- Nash equilibrium loses stability and bounded but perpetual oscillations arise already for a triopoly. For more than three rms oscillations grow unbounded, but they are limited once the non-negativity price and demand constraints bind. Whereas Theocharis (960) focused only on the homogeneous (Cournot) adjustment process, more recent research etends to models of heterogeneous epectations 2. For instance, Hommes, Ochea, and Tuinstra (208) introduced a framework in which these heuristics compete in a quantity-setting oligopoly with arbitrary number of rms. Each rm chooses a behavioural rule from a nite set of di erent rules, which are assumed to be commonly known. When making a choice concerning the behavioural rules, a rm takes the past performance of the rules, i.e., the past realized pro t net of the cost associated with the behavioural rules to compare tness. Both past performance and costs associated with the behavioural rules are publicly available. This implies that successful heuristics will continue to be used, while unsuccessful behavioural rules are dropped. This strategic behaviour thus causes the distribution of fractions of rms over a given set of behavioural rules to change per period. Hommes, Ochea, and Tuinstra (208) focus on the Cournot heuristic in competition with the Nash quantity or with rational rms. Interestingly Huck, Normann, and Oechssler (2002) discuss a linear Cournot oligopoly eperiment with four rms. They do not nd that quantities eplode as the Theocharis (960) model predicts, instead the time average quantities converge to the Cournot-Nash equilibrium quantity, although there is substantial volatility around the Cournot-Nash equilibrium quantity. Firms that display Cournot behaviour take the current period s aggregate output of their competitors as a predictor for the net period competitors aggregate output and best-respond to that. 2 In models with heterogeneous epectations producers can have di erent heuristics to adjust their production. 2

4 There is a growing interest, both theoretical and eperimental, in the study of the performance of imitative players in various classes of games. Schipper (2009) investigates imitate-thebest players and optimizers in Cournot oligopoly and nds that in the long-run, stationary distribution of the stochastic process imitators are better o. Moreover imitation can be unbeatable if imitate-the-best heuristic is not subjected to a money pump, i.e. game is not of Rock-Scissors-Paper variety (Duersch, Oechssler, and Schipper (202)). Subsequently, Duersch, Oechssler, and Schipper (204) show that unconditional imitation (of the tit-for-tat variety) is essentially unbeatable in class of potential games. Huck, Normann, and Oechssler (2002) nd that a process where participants mi between the Cournot adjustment heuristic an imitating the previous period s average quantity gives the best description of behaviour. Duersch, Kolb, Oechssler, and Schipper (200) analyse a Cournot duopoly, subjects earn on average higher pro ts when playing against "best-response" computers than against "imitate" computers. Therefore we focus on competition of the imitation heuristic with the Cournot heuristic. Moreover, since classical economic theory assumes rationality, we investigate the dynamics in competition with this heuristic too. In total ve models where imitators compete with Cournot and/or rational rms are investigated analytically. The framework created by Hommes, Ochea, and Tuinstra (208) will be followed in order to do the analytics. Our concern is, rst of all, under what circumstances rms may want to switch between behavioural rules over time and second, once the Cournot-Nash equilibrium is reached whether all rms will keep producing the Cournot-Nash quantity or deviate. Main ndings are that, (i) in the case when Cournot rms compete with imitators that the threshold on the number of rms that changes the system from stable to unstable is 7, (ii) when rational rms compete with imitators, in the speci c scenario of linear inverse demand and constant marginal cost, the system is always stable regardless of the game and behavioural parameters, (iii) in the case when rational rms, Cournot rms and imitators compete, the stability depends on the evolutionary pressure and the the stability of the 3

5 cheapest heuristic(s). When the cheapest behavioural rule is stable, the dynamics converge to a situation where most rms use this behavioural rule and all rms produce the Cournot-Nash equilibrium quantity. So having more information about the market does not necessarily lead to higher pro ts due to information costs. In the case when the cheapest heuristic is unstable, complicated endogenous uctuations may occur. In particular, when the evolutionary pressure is high or when the number of rms passes a certain threshold. Note that the nonlinearity causing this erratic behaviour comes from the endogenously updating of the fractions, because in our leading eample the speci cations were linear. The remainder of this paper is organized as follows, in Section 2 the theoretical framework is introduced, here the quantity and population dynamics will be eplained etensively. In Section 3 the dynamics will be investigated under eogenous population dynamics whereas in Section 4 the stability of the system will be investigated under endogenous population dynamics. In the fth Section the results of section four are combined and the stability of a system where rational, Cournot and imitators compete in one economy under endogenous fraction dynamics is investigated. Finally, we conclude in Section 6. 2 Theoretical Framework Consider a nite population of rms who are competing on the market for a certain good, each discrete-time period all producers have to decide their production plans for the net period. However, instead of simultaneously choosing the supplied quantities directly, the rms act according to behavioural rules that eactly prescribe the quantity to be supplied. Before the evolutionary model is studied a brief review of the traditional, static Cournot model will be given. Consider a symmetric Cournot oligopoly game, where q i denotes the quantity supplied by rm i; where i = ; :::n. Net to that let Q = P n i= q i be the aggregated production. Furthermore let P (Q) denote the twice di erentiable, nonnegative and non-increasing inverse 4

6 demand function and let C(q i ) denote the twice di erentiable non-decreasing cost function, which is the same for all rms. For rm i the resulting pro t function from the above described model is given by i (q i ; Q i ) = P (q i + Q i )q i C(q i ); i = ; :::n () where Q i = P j6=i q j. Assume that the pro t function of a rm is strictly concave in its own output q i. The pro t maimizing strategy of rm i, taking the quantity supplied by the competitors as given, results in the well-known best-reply function for rm i, which is given by q i = R i (Q i ) = Argma[P (q i + Q i )q i C(q i )]: q i Due to symmetry, all rms have the same best-reply function R(). Moreover, the symmetric Cournot-Nash equilibrium quantity q corresponds to the solution of q = R((n )q ): Strict concavity of the pro t function ensures that such a Cournot-Nash equilibrium eists. For simplicity assume that q is the unique symmetric Cournot-Nash equilibrium strategy. 3 In this thesis focus lays on the following speci cation of the Cournot oligopoly game which will be called the leading eample. This is the original speci cation Theocharis (960) used, where inverse demand is linear and marginal costs are constant. The inverse demand and cost function are given by P (q i + Q i ) = a b(q i + Q i ) and C(q i ) = cq i ; i = ; :::n respectively. First, in order to have a strictly concave pro t function assume that b > 0. Furthermore, for strictly positive prices assume that Q < a. For these speci cations of the b 3 The Cournot duopoly game may also have asymmetric Cournot-Nash equilibria, but they do not correspond to equilibria of the evolutionary game when there is a single population. For the linear-quadratic speci cation of the Cournot oligopoly model speci ed below, there can indeed be asymmetric boundary equilibria, but they do not in uence the dynamics of the evolutionary model. 5

7 inverse demand function and cost function the reaction function is given by q i = R(Q i ) = a c 2b 2 Q i = q 2 (Q i (n )q ): (2) Note that if the other rms produce on average more (less) than the Cournot-Nash equilibrium quantity, rm i reacts by producing less (more) than that quantity. Straightforward calculations show that in this case the Cournot-Nash equilibrium quantity, aggregated production, price and pro t are equal to q = a c, b(n+) Q = a P = a+nc n+ and = (q i ; Q i) = (a c)2 b(n+) 2. c n, b n+ Traditional Cournot analysis refers to a static environment. However, in a dynamic setting the reaction function introduced above can be used to study the so called Cournot-dynamics where rms best-reply to their epectations q i;t = R(Q e i;t); i = ; :::n where q i;t denotes the quantity supplied by player i in period t. The symmetric Cournot-Nash equilibrium where all rms produce q is stable under the Cournot-dynamics if (n )jr 0 )j <. Main interest is on how rm i decides to play q and on top of that, what does rm i believe about Q i when the production decision has to be made. In the net Subsection the description of the quantity dynamics will be given. In Subsection 2.2 some local instability results for the general evolutionary system are discussed. In Subsection 2.3 the population dynamics will be discussed. 2. Production plans In the Cournot oligopoly game the producers have to form epectations about opponents production plans. Based on this epectation rms decide how much to produce the net 6

8 period. One approach is to assume complete information, i.e. rational rms with common knowledge of rationality. This implies that rms have perfect foresight about competitors aggregated production plan, i.e. Q e i;t+ = Q i;t+ : This results in the following production plan: q i;t+ = R(Q i;t+ ); i = ; :::n Alternatively one may consider rules that require less information, for eample Q e i;t+ = Q i;t : This results in the following production plan: q i;t+ = R(Q i;t ); i = ; :::n (3) where rms epect that aggregated production in the net period equals current aggregated production. This is the so called Cournot adjustment heuristic. It is a broadly supported idea that not all producers best-reply to their epectations. Eperiments (Huck, Normann, and Oechssler (2002)) show that people often imitate others behaviour. A heuristic that possibly seizes this production plan is the so called imitationheuristic. Imitators belief that everyone else can t be wrong and will therefore produce the average of the other players production in the net period, i.e. q i;t+ = Q i;t ; i = ; :::; n: (4) n Finally, Bosch-Domènech and Vriend (2003) test the importance of models of behaviour characterised by imitation of successful behaviour, that is to imitate the quantity which the rm with the highest pro t in the current period produced, i.e. q i;t+ = q j;t ; i = ; :::n; where j;t = Maf ;t ; :::; k;t g: They nd that the players do not rely more on imitation of successful behaviour in more demanding environments and eplain the di erent output decisions as predominantly relate 7

9 to a general disorientation of the players, and more speci cally to a signi cant decrease of best responses. In the net subsection we will investigate the dynamics under epectation rule (3) and (4) in greater detail. 2.2 Instability threshold 2.2. Cournot adjustment heuristic If all rms use the Cournot adjustment heuristic (3), quantities evolve according to the following system of n rst order di erence equations q ;t+ = R(q 2;t + q 3;t + : : : + q n;t ); q 2;t+ = R(q ;t + q 3;t + : : : + q n;t ); = (5) q n;t+ = R(q 2;t + q 3;t + : : : + q n;t ): Local stability of the Cournot-Nash equilibrium depends on the eigenvalues of the Jacobian matri J of the system of equations (5), evaluated at that Cournot-Nash equilibrium q. This Jacobian matri is given by 0 0 R 0 (Q ) R 0 (Q ) R 0 (Q 2) 0. Jj q =. : (6) B... R 0 Q (n ) A R 0 (Q n) R 0 (Q n) 0 Firms do not respond to their own previous production, therefore all diagonal elements are equal to zero. All o -diagonal elements in row i are equal to R 0 (Q i), since individual production levels only enter through aggregate production of the other rms. Moreover, at the 8

10 symmetric Cournot-Nash equilibrium we have Q i = (n )q for i = ; :::; n, thus all o diagonal elements of (6) are equal to R 0 ). The Jacobian matri (6) thus has n eigenvalues equal to R 0 ) and one eigenvalue equal to (n )R 0 ), which is the largest in absolute value. From this it follows directly that the symmetric Cournot-Nash equilibrium is stable whenever (n) (n )jr 0 )j < ; (7) where (n) is de ned as the largest eigenvalue of the Jacobian, evaluated at the equilibrium. Leading eample. From equation (2) it can easily be seen that R 0 (Q i) = 2, meaning that if others aggregated output increases by one unit, the Cournot-Nash rms decrease their output by 2 units. From stability condition (7) it follows that the Cournot-Nash equilibrium is stable for this speci cation only when n = 2 and unstable when n > 3 (and neutrally stable, resulting in bounded oscillations, for n = 3). The reason for this instability is overshooting : if aggregated output is above (below) the Cournot-Nash equilibrium quantity, rms react by reducing (increasing) their output. For n > 3 this aggregated reduction (increase) in output is so large that the resulting deviation of aggregated output from the equilibrium quantity is larger in the net period than in the current, and so on Imitation heuristic If all rms use the imitation heuristic (4), quantities evolve according to the following system of n equations q ;t+ = Q ;t n ; q 2;t+ = Q 2;t n ; = (8) q n;t+ = Q n;t n : 9

11 Local stability of the Cournot-Nash equilibrium with only imitation rms depends on the eigenvalues of the Jacobian matri of the system of equations (8) evaluated at that Cournot- Nash equilibrium q. This Jacobian matri is given by 0 Jj q = 0 n 0 n. n.... n 0 n n : (9) C A Imitators only respond to other rms production and do not respond to their own production, therefore all diagonal elements are equal to zero. If one competitor increases current production by one unit, an imitator will increase net production with n unit, therefore all o -diagonal elements are equal to. The Jacobian matri (6) thus has n eigenvalues n equal to n and one eigenvalue equal to (n ) n = which is the largest in absolute value. Therefore it follows immediately that the Cournot-Nash equilibrium is neutrally stable independent of n and system structure (price and cost function). The reason for this is that if one producer changes his production plan the economy will stabilize to a new equilibrium unequal to q and will remain at this new equilibrium until one producer deviates again. In fact this system has in nitely many neutrally stable equilibria, namely if q i = q y 8i; the system is neutrally stable for all q y. 2.3 Population dynamics In the previous sections it is eplained how the supplied quantities evolve over time under the Cournot and the imitation heuristic. In this section it will be eplained how the population fractions evolve over time. Let us rst introduce the vector t which has entries equal to k;t, which is the fraction of the population that uses heuristic k at time t. Thus for every time t, t denotes the K-dimensional vector of fractions for each strategy/heuristic and belongs to the K-dimensional simple K = f t 2 R K : P K k= k;t = ; 0 k;t 8kg. We will now 0

12 describe how the fractions k;t evolve over time. It is assumed that the choice of a behavioural rule is based on its past performance, capturing the idea that more successful rules will be used more frequently. Evolutionary game theory deals with games played within a (large) population over a long time horizon. Its main ingredients are its underlying game, in this thesis the Cournot one-shot game, and the evolutionary dynamic class which de nes a dynamical system on the state of the population. The evolutionary dynamical system depends on current fractions t and current tness U t. In general, such an evolutionary dynamic in discrete time, describing how the population fractions evolve, is given by k;t+ = K(U t ; t ) (0) with U t = (U ;t ; :::; U K;t ) 0 the vector of average utilities and t = ( ;t ; :::; K;t ) 0 the factor of fractions. To make sure that the population dynamics is well-behaved in terms of dynamic implications we assume that K(; ) is continuous, nondecreasing in U k;t, and such that the population state remains in the K-dimensional unit simple K. In the net Subsection leading class of population dynamics will be eplained in detail, the Logit evolutionary dynamics Discrete choice models - the Logit evolutionary dynamics The Logit evolutionary dynamic is treated etensively in Brock and Hommes (997). This Section contains a brief discussion. In order to update the fractions we assume that average utility of all heuristics is publicly observable. Suppose that the observed average utility associated behavioural rule H k takes the form ~U k = U k + k; where k s are IID. This captures the idea of bounded rationality since individuals do not

13 necessarily select the rule that yields the highest utility. The parameter represents the evolutionary pressure. Notice that in the etreme case where = 0 we have completely random behaviour: the noise is so large that observed average utility is equal for all behavioural rules. Each behavioural rule is thus chosen with equal probability: k;t = K 8k. In the other etreme case, when! obscures and everybody switches to the most pro table strategy each period. If the noise terms k s are distributed according to the etreme value distribution the evolutionary fraction dynamic results in the so-called multinomial Logit evolutionary dynamic, the following updating dynamic is given by k;t+ = eu k;t KX j= e U j;t ; k = ; :::; K: () The equilibrium fractions are given by T k;t+ = e( k ) KX j= e ( T j ) ; k = ; :::; K (2) In case of equal costs of the heuristics, equilibrium fractions are thus given by k = K 8k, since production is equal and thus pro ts are equal. Note that the population dynamics remains in the interior of the unit simple for nite. This implies that in each time period all behaviour rules are present in the population and no behavioural rule will ever vanish (this is the so-called no-etinction condition). Furthermore, no new behavioural rules emerge from this model (this is the so-called no-creation condition). In the leading eamples we will focus on the Logit evolutionary dynamics. First of all because this dynamic is also used in Hommes, Ochea, and Tuinstra (208) and therefore creates the possibility to make a good comparison and furthermore because the Logit evolutionary dynamic has by de nition nice regularity/continuity conditions (0 k ). 2

14 3 Heterogeneity in behaviour in Cournot oligopolies In this Section we study the Cournot game and introduce heterogeneity in production plans. In this Section we focus on competition between two heuristics. We rela this in Section ve, where we study the competition between rational, Cournot and imitation rms. First we study competition between the Cournot and the imitation rms, with this as an eample, two theories will be presented on how to model this heterogeneity in production. In the rst theory the rms select their heuristic that completely describes how much to supply in the net period. They select heuristic k with probability k. In the second theory n rms are randomly picked from a large population of rms in which a fraction k plays according to strategy k. Main di erence is that the rms observe under the second theory more outcomes and thus under the law of large numbers lets the production plans within a heuristic converge whereas in the rst theory all rms (even the rms using the same heuristic) have di erent production plans, making the dynamics analytically untractable. After this etensive study of competition between Cournot rms and imitators, we introduce another model where rational rms compete with imitation rms. Since the dynamics are only tractable under theory 2, we will focus on this theory when studying this model. The assumption of ed for each period will be relaed in section Cournot vs. Imitation rms 3.. Theory 2: A large population game In order to facilitate studying the aggregate behaviour of a heterogeneous set of interacting quantity-setting-heuristics we study the Cournot model as a population game. Consider a large population of rms from which in each period groups of n rms are sampled randomly and matched to play the one-shot n-player Cournot game. We assume that a ed fraction of of the large population of rms uses the Cournot heuristic and the others use the imitation heuristic. After each one-shot Cournot game, the random matching procedure is repeated, 3

15 leading to new combinations types of rms. The distribution of possible samples follows a binomial distribution with parameters n; and. Below the eample Cournot vs. Imitation rms will be discussed again but now under theory 2 of random matching. Suppose that a fraction of of the population of the rms uses the Cournot heuristic and observes the population-wide average quantity q t and best responds to it, qt+ C = R((n )q t ), where q C t is the quantity produced by each Cournot rm in period t. Consequently a fraction of rms of the large population makes use of the the imitation heuristic. Making use of the law of large numbers, the average quantity played in period t can be epressed as q t = q C t + ( )q I t : Remember that imitation rms produce in the net period the by the other rms average produced quantity in the current period qi;t+ I = Q i;t. Again by a law of large numbers we n obtain Q i;t n! q twhen n!. Therefore we obtain the following quantity dynamics q C t+ = R((n )(q C t + ( )q I t )) q I t+ = q C t + ( )q I t : (3) Note that this is a 2-dimensional dynamical system which dimension cannot be reduced. Furthermore the Cournot-Nash equilibrium is not the unique equilibrium of the imitation rule, in fact all quantities are. The Cournot-Nash equilibrium is, however, still the unique equilibrium quantity of the complete dynamical system. Proposition The Cournot-Nash equilibrium,where all rms produce the Cournot-Nash quantity (q ; q ), is a locally stable ed point for the model with eogenous fractions of Cournot and imitation rms if and only if j + (n )(R 0 )j < : (4) 4

16 Proof. It can easily be shown that the Jacobian matri, evaluated at the Cournot-Nash equilibrium (q ; q ), is given by 0 (n )R 0 ) (n )( )R 0 ) B A : (5) The corresponding eigenvalues are = 0 and 2 = + (n )(R 0 ). Here 2 is the largest eigenvalue in absolute value. Thus the system is stable if j 2 j <, this is the condition stated in the proposition. Leading eample. Here R 0 ) = 2 substituting this in equation (4) gives, after some simpli cation n < 4 : (6) Meaning that an economy with as much Cournot rms as imitators( = ) is stable if n < 7. 2 Net to that as found earlier, an economy with only cournot rms ( = ) is stable if n < 3. Furthermore, an economy where close to all rms use the imitation heuristic, but some Cournot rms eist ( close to zero), the economy is always stable. 3.2 Rational vs. Imitation rms In this section we focus on the dynamics when there is competition between rational and imitation rms. Remember that we will model this heterogeneity under theory 2 since this makes the dynamics analytically tractable. We set the fraction of rational rms equal to. A fully rational rm is assumed to know the fraction of imitation rms. Moreover, it knows eactly how much all rms will produce. However, we assume that it does not know the composition of rms in its market (or has to make a production decision before observing 5

17 this). The rational quantity dynamics therefore have the following structure q R = ArgmaE[P (q i + Q i )q i C(q i )]: q i It forms epectations over all possible mitures of heuristics resulting from randomly drawing n other players from a large population, of which each with chance is a rational rm too, and with chance is an imitator. Rational rm i therefore chooses quantity q i such that his objective function, its own epected utility Xn n Ut R (q i;t jqt R ; qt I ; ) = k ( ) n k [P ((n k)qt I +kqt R +q t;i )q t;i C(q t;i )]; (7) k k=0 is maimized given the production of the other players and the population fractions. Here qt R is the symmetric output level of all of the other rational rms in period t, and qt I is the output level of all of the imitation rms. The rst order condition for an optimum is characterized by equality between marginal cost an epected marginal revenue. Typically, marginal revenue in the realized market will di er from marginal costs. Given the value of qt I and the fraction, all rational rms coordinate on the same output level qt R. This gives the rst order condition U R t (q i;t jq R t ; q I t ; ) q i;t = 0; which equals to: Xn n k ( ) n k k k=0 [P ((n k)q I t +(k + )q R t ) + q R t P 0 ((n k)q I t + (k + )q R t ) C 0 (q R t )] = 0: (8) Let the solution to equation (8) be given by q R t = H R (q I t ; ), the full system of equations is 6

18 thus given by q R t+ = H R (q I t+; ) = H R (q R t + ( )q I t ; ) q I t+ = q R t + ( )q I t : (9) It is easily checked that if the imitators play the Cournot-Nash equilibrium quantity q, or if all rms are rational, the rational rms will play the Cournot-Nash equilibrium quantity, that is H R (q ; ) = q, for all and H R (q I ; ) = q for all q I. Moreover, if a rational rm is certain it will only meet imitation rms (that is = 0), it plays a best response to the currently average played quantity, that is H R (q I t ; 0) = R((n )q I t ), for all q I t. In the remainder we will denote the partial derivatives of H R (q; ) with respect to q and by H R q (q; ) and H R (q; ) respectively. Proposition 2 The Cournot-Nash equilibrium, where all rms produce the Cournot-Nash quantity (q ; q ), is a locally stable ed point for the model with eogenous fractions of rational and imitation rms if and only if jh q (q; ) + j < (20) Proof. In order to determine the local stability of the equilibrium (q ; q ) where all rms produce the Cournot-Nash quantity, we need to determine the eigenvalues of the Jacobian matri of system (9), evaluated at the equilibrium. It can be shown that this Jacobian matri is given by 0 B Jj q ;q H q(q ; ) ( )H q (q ; ) C A ; (2) which has eigenvalues = H q (q ; ) + and 2 = 0. Consequently the system is locally stable when j j <, this is eactly the condition stated. Leading eample. In the leading eample the implicit function de ning q R t (Eq. (8)) when 7

19 absolute Largest eigenvalue using that Xn n k ( ) n k = and k k=0 Xn n k k=0 k ( ) n k k = (n ) boils down to q R t+ = H R (q I t+; ) = a c b(2 + (n )) (n )( ) 2 + (n ) (qr t + ( )q I t ): The system of equations for the leading eample is given by q R t+ = H R (q I t+; ) = a c b(2 + (n )) (n )( ) 2 + (n ) (qr t + ( )q I t ) (22) q I t+ = q R t + ( )q I t The eigenvalues of this system are given by = 0 and 2 = system is stable if j 2 j <. Since 0 and the economy is always stable in the linear speci cation. In Figure this is graphically shown. (n )( ), thus the 2+(n ) (n )( ) 2+(n ) < ; this stability condition always holds Firms Cournot firms Figure : Largest eigenvalue for the model rational vs. imitation rms. The largest eigenvalue decreases when the number of rms increases and when the fraction of rational players increases. Since an economy consisting of only imitation rms is neutrally stable, this model is stable for all combinations of and n. 8

20 4 Evolutionary competition between two heuristics In this Section we develop an evolutionary version of the model outlined in Section 3, i.e. relaing the assumption that is ed. As before in ever period t, n rms play the n-player Cournot game. We now assume that the fractions of rms using a heuristic evolves over time according to a general monotone selection dynamic, capturing the idea that heuristics that perform relatively better are more likely to spread through the population as eplained in Section 2.3, Eq. (0), here it is eplained that future fractions depend on current fractions and current tness. Under the assumption of random interactions, the tness of heuristic k is determined by averaging the payo s from from each interaction with weights given by the chance of that speci c state minus the information cost of using the heuristic. Denoting with t the epected payo vector in period t, its entries - individual payo or tness in biological terms - of strategy is given by: Xn k=0 ;t = F (q ;t ; q 2;t ; t ) = (n )! k!(n k)! k t ( t ) n k P ((k + )q ;t + (n k)q 2;t )q ;t C(q ;t ); (23) and with epected pro ts for heuristic 2 given by 2 = F (q 2 ; q ; ). If the population of rms and the number of groups of n rms drawn from that population are large enough, average pro ts will be approimated well by these epected pro ts, which we will use therefore as a proy for average pro ts from now on. There might be a substantial di erence in sophistication between di erent heuristics. As a consequence some heuristics may require more information or e ort to implement than others. Therefore we allow for the possibility that heuristics involve information cost C k 0, that may di er across heuristics. Fitness of a heuristic is then given by the average pro ts generated in the game minus the information costs, U k = k C k. We only use the realized 9

21 pro t to determine the tness measure of a behavioural rule. The tness measure can be generalized by weighting the utility of the past M periods, yielding similar results (Tuinstra (999)). We assume that the above tness measures U k are publicly observable. Having the tness measure we are ready to introduce the population dynamics. Let the fraction of rms using the rst heuristic be given by in period t. This fraction evolves endogenously according to an evolutionary dynamic which is an increasing function in the di erence between the current tness of the two heuristics and current fraction, that is t+ = K(U ;t U 2;t ) = K(U ;t ): The map K : R! [0; ] is a continuously di erentiable, monotonically increasing function with K(0) =, K() + K( ) =, meaning that it is symmetric around = 0, 2 lim! K() = 0 and lim! K() = In the following two sections we will derive two dynamical versions of the two models discussed in Section 3 and investigate their stability. First we investigate the stability of the Cournot-Nash equilibrium for the model with endogenous fractions of Cournot and imitation rms and second we investigate the stability of the Cournot-Nash equilibrium for the model with endogenous fractions of rational and imitation rms. 4. Cournot versus Imitation rms The dynamics in this section consists of three equations, two equations describing the quantity dynamics: the production of the Cournot rms and the production of the imitation rms. Net to that we need one equation to describe the dynamics of the population fraction. The population and quantity dynamics look like the following system of three equations: 20

22 q C t+ = R((n )( t q C t + ( t )q I t )); q I t+ = t q C t + ( t )q I t (24) t+ = K(U t ); where U t = U C;t U I;t : Note that this is a 3-dimensional dynamical system which dimensions cannot be reduced. Furthermore, the Cournot-Nash equilibrium quantity q is the unique equilibrium quantity of the complete dynamical system. Let be the unique equilibrium fraction such that = K( C). Without specializing the population dynamics K() we have the result as stated in the proposition below. Proposition 3 The Cournot-Nash equilibrium (q ; q ; ) is a locally stable ed point for the model with endogenous fractions of Cournot and imitators where all rms produce the Cournot-Nash quantity, rms if and only if R((n )q )(n ) > 2: (25) Proof. It can easily be shown that the Jacobian matri of system 24, evaluated at the equilibrium (q ; q ; ) is given by 0 (n ) R 0 ) (n )( )R 0 ) 0 Jj q ;q ; = B 0 K(U J 3 J t) q A : (26) 32 t ;q ; The eigenvalues of this Jacobian matri are, independently of J 3 and J 32 given by = R((n )q )(n ) + ; 2 = K(U t) t and 3 = 0: (27) q ;q ; 2

23 To our best knowledge of possible population dynamics K(Ut) t is positive but smaller than. This holds for all population dynamics discussed in Section 2.3. Therefore, for the system to be stable we need R((n )q )(n ) > 2; which is eactly the condition stated in the proposition. Note that this is the same condition we derived in Section 3.. where we ed. Leading eample. In the equilibrium, when all rms produce the same quantity, pro ts are equal and therefore the equilibrium fraction simpli es to = K( C). The equilibrium quantities are given by q. Here R 0 ) =, lling this in equation (25) gives the stability 2 condition for the leading eample. Thus the equilibrium (q ; q ; ) is stable when n < 4 In Figure 2 the model is simulated under Logit-dynamics with intensity of choice parameter, see Brock and Hommes (997). Panel (a) depicts a period-doubling route to chaotic quantity dynamics as the number of rms n increases. The rst period-doubling bifurcation is for n = 7 as calculated analytically. Panel (b) displays oscillating time series of produced quantity by the Cournot and imitation rms and the equilibrium quantity fraction q. As one can see the Cournot quantities are uctuating more than the imitation quantities. The stabilizing e ect of the imitation rms is here clearly visible, when Cournot rms produce more (less) then the Cournot-Nash equilibrium quantity, the imitation rms produce less (more) than the Cournot-Nash equilibrium quantity and therefore decrease the aggregated deviation from the equilibrium. Panel (c) displays the resulting Cournot pro t di erential. C I. Panel (d) displays the resulting oscillating time series of the Cournot and imitation fractions. In Panel (e) a phase portrait is shown for the Cournot heuristic whereas in Panel (f) a phase portrait for the imitation heuristic is shown. In Panel (g) the largest Lyapunov eponent for an increasing number of rms is shown. Game and behavioural parameters are equal set to: n = 0, a = 7, b =, c =, C C = 0, C I = 0, = 0:05. Initial conditions are set equal to: q0 C = 0:8, q0 I = 0:8, 0 = 0:5 When the evolutionary pressure increases, the system evolves to an equilibrium di erent from the Cournot-Nash equilibrium where the 22

24 imitation rms produce more than the Cournot-Nash equilibrium whereas the Cournot rms produce less. Imitation pro ts are therefore much higher and as a consequence the complete population switches to the imitation heuristic. (a) Bifurcation diagram (q t ; n) (b) Time path of Cournot and imitiation quantities (c) Cournot pro t di erential (d) Time path Cournot fraction (e) Cournot phase plot (f) Imitation phase plot Figure 2: Linear n-player Cournot game with endogenous fraction dynamics. The bifurcation diagram is re-plotted in Fig. 3 under the same game and behavioural parameters and initial conditions, the only di erence is that now = 3. 23

25 Figure 3: Bifurcation diagram (q t ; n) with = 3 When :7 < n < 2:8 the imitation rms produce more then the Cournot-Nash equilibrium quantity while the Cournot rms produce less. This results in higher pro ts for the imitators and therefore the complete populations switches to imitators ( = 0). When 2:8 n 3:2 all rms produce the Cournot-Nash equilibrium quantity again, therefore pro ts and thus fractions are equal. When n > 3:2 The imitation rms produce again more then the equilibrium quantity while the Cournot rms produce less, eept when n is close to 3.65, then all rms produce the Cournot-Nash equilibrium quantity. Finally, when n > 5:6 the imitation rms produce so much that the Cournot rms decide to produce nothing (q C = 0). 4.2 Rational vs. Imitation rms As in the previous Section we need a 3-dimensional system to describe the dynamics of the model. The rational rms produce each period such that their epected pro t is maimized whereas an imitator produces in the net period the currently average played quantity. The rational quantity dynamics therefore have the following structure q R t = ArgmaE[P (q i;t + Q i;t )q i C(q i;t )]: q i It forms epectations over all possible mitures of heuristics resulting from randomly drawing 24

26 n other players from a large population, of which each with chance t is a rational rm too, and with chance t is a imitator. Rational rm i therefore chooses quantity q i such that his objective function, its own epected utility Xn n Ut R (q i;t jqt R ; qt I ; t ) = t k ( t ) n k [P ((n k)qt I +kqt R +q t;i )q t;i C(q t;i )]; (28) k k=0 is maimized given the production of the other players and the population fraction. Here q R t is the symmetric output level of each of the other rational rms in period t, and q I t is the output level of each of the imitator rms in period t. The rst order condition for an optimum is characterized by equality between marginal cost an epected marginal revenue. Given the value of qt I and the fraction t, all rational rms coordinate on the same output level qt R. This gives the rst order condition U R t (q i;t jq R t ; q I t ; t ) q i;t = 0; which equals to Xn n t k ( t ) n k k k=0 [P ((n k)q I t +(k + )q R t ) + q R t P 0 ((n k)q I t + (k + )q R t ) C 0 (q R t )] = 0: (29) Let the solution to equation (29) be given by q R t = H R (q I t ; t ), the full system of equations is thus given by q R t+ = H R (q I t+; t+ ) q I t+ = t q R t + ( t )q I t (30) t+ = K(U t ): where U t = U R t U I t. It is easily checked that if the imitators play the Cournot-Nash 25

27 equilibrium quantity q, or if all rms are rational, then the rational rms will play the Cournot-Nash equilibrium quantity, that is H R (q ; ) = q, for all and H R (q I ; ) = q for all q I. Moreover, if a rational rm is certain it will only meet imitation rms (that is = 0), it plays a best response to the currently average played quantity, that is H R (qt I ; 0) = R((n )qt I ), for all qt I. In the remainder we will denote the partial derivatives of H R (q; ) with respect to q and by Hq R (q; ) and H R (q; ) respectively. Proposition 4 The Cournot-Nash equilibrium (q ; q ; ) is a locally stable ed point for the model with endogenous fractions of rational and imitation rms, where all rms produce the Cournot-Nash quantity, if and only if j H q (q ; ) + j < (3) Proof. Since a dynamical system can only depend on lagged variables, we substitute the second and third equation into the rst. This gives us the following system that depends only on lagged variables. q R t+ = = H R ( t q R t + ( t )q I t ; K(U R t )) q I t+ = 2 = t q R t + ( t )q I t (32) t+ = 3 = K(U R t ): In the equilibrium all rms produce the Cournot-Nash quantity q, therefore pro ts are equal, hence the equilibrium fraction is given by = K( C). In order to determine the local stability of the equilibrium (q ; q ; ) where all rms produce the Cournot-Nash quantity, we need to determine the eigenvalues of the Jacobian matri of system (9), evaluated at the equilibrium. The partial derivatives of 2 with respect to q R t, q I t and t, evaluated at the equilibrium are, and 0 respectively. 26

28 Net, let us determine the partial derivatives of 3 with respect to q R t, q I t and t, respectively. To that end, note that we can write the pro t di erential as Xn Ut R = R t I t C = A k ( t )D k (qt R ; qt I ; t ) C; with A k ( t ) = n k k t ( t ) n k, which does not depend upon q R and q I, and k=0 D k (q R t ; q I t ; t ) =P ((k + )q R + (n k)q I )q R C(q R ) [P (kq R + (n k)q I )q I C(q I )]; (33) which depends upon t through q R t = H(q I t ; t ). Note that D k (q R t ; q ; t ) = 0, moreover the partial derivatives of D k (q R t ; q I t ; t ), evaluated at the equilibrium (q ; q ; ) are given by D k (q R t ; q I t ; t ) q R t D k (q R t ; q I t ; t ) q I t D k (q R t ; q I t ; t ) t = [P 0 )q + P (Q ) C 0 )]H q (q ; ) = 0; (q ;q ; ) = [P 0 )q P (Q ) + C 0 )] = 0; (q ;q ; ) = [P 0 )q P (Q ) C 0 )]H (q ; ) = 0: (q ;q ; ) The second equalities follows from the fact that P 0 )q + P (Q ) C 0 ) = 0 is the rst order condition of any rm in a Cournot-Nash equilibrium. Furthermore D k (q ; q ; ) = 0 for all by the rst order condition for a Cournot-Nash equilibrium. Using this it follows immediately that: 3 q R t (q ;q ; ) = K 0 ( C) U t q R t k=0 (q ;q ; ) Xn = K 0 ( C) A k ( ) D k(qt R ; qt I ; t ) qt R = 0 (q ;q ; ) (34) 27

29 and 3 q I t (q ;q ; ) = K 0 ( C) U t q I t k=0 (q ;q ; ) Xn = K 0 ( C) A k ( ) D k(qt R ; qt I ; t ) qt I = 0 (q ;q ; ) (35) and 3 t (q ;q ; ) = K 0 ( C) U t t k=0 (q ;q ; ) Xn = K 0 ( C) [A k ( ) D k(qt R ; qt I ; t ) t (q ;q ; ) + A k() D k (q R ; qt I ; )] t (36) = 0: This leaves us to eamine the partial derivatives of with respect to q R t, q I t and t, evaluated at the equilibrium. q R t (q ;q ; ) = Hq R (q ; ) + K(U t) H qt R (q ; ) = H R q (q ; ) (37) and q I t (q ;q ; ) = ( )Hq R (q ; ) + K(U t) H qt I (q ; ) = ( )H R q (q ; ) (38) and 28

30 t (q ;q ; ) = (q q )H R q (q ; ) + K(U t) t H (q ; ) = 0 (39) Therefore the Jacobian matri, evaluated at the equilibrium is given by 0 Hq R (q ; ) ( )Hq R (q ; ) 0 Jj q ;q ; = B 0 A : (40) Which has eigenvalues = H q (q; ) +, 2 = 0 and 3 = 0. Consequently the system is locally stable when j j <, this is eactly the condition stated in proposition 4. Note again the similarity with the condition in Section 3 where we ed the fraction. Leading eample. Since the stability condition is the similar to the condition derived in Section 3.2, the equilibrium (q ; q ; ) is stable for all n in this linear speci cation. 5 Rational vs. Cournot vs. Imitation In this section we combine the ideas that we gathered in Section 4. We will investigate the dynamics when the three heuristics discussed before compete. As before every round n rms are drawn from a large pool of rms to play the one-shot Cournot game. From this large pool of rms a fraction t R plays according to the rational strategy in period t, a fraction t C plays according to the Cournot heuristic in period t and consequently the fraction of imitators in period t is determined by t R t C. As in Section 4 the tness of a heuristic is determined by the average payo minus the information cost of using that heuristic. Again the average pro ts will be approimated by the epected pro ts but in contrast to Section 4 the distribution of states now follows a multinomial distribution instead of a binomial distribution. In general 29

31 the average pro t of a rm producing q and competing with other rms that produce either q 2 or q 3 given the fractions and 2 is stated below, in this average pro t approimation the pro t in each state is weighted by the chance of this state. X ;t = F (q ;t ; q 2;t ; q 3;t ; ;t ; 2;t ) = (n )! k!k 2!(n k k 2 )! k ;t k 2 2;t( ;t 2;t ) n k k 2 (4) P ((k + )q ;t + k 2 q 2;t + (n k k 2 )q 3;t )q ;t C(q ;t ); The summation is over all possible combinations of k and k 2, which stand for the number of other rms producing q and q 2 respectively, that is: = fk ; k 2 2 I 2 : 0 k n ; 0 k 2 n ; 0 k + k 2 n g: Epected pro ts for heuristic 2 in period t are given by F (q 2;t ; q ;t ; q 3;t ; 2;t ; ;t ), epected pro ts for heuristic 3 in period t are given by F (q 3;t ; q 2;t ; q ;t ; ;t 2;t ; 2;t ). The complete dynamical system consists of ve equations, three for the quantity dynamics and two to describe how the fractions evolve. As in all previous sections, the Cournot rms play in the net period a best-response to the current aggregated output of the others, imitators play in the net period the average produced quantity by the others in the current period. Rational players produce every period the quantity that maimizes epected payo given the fractions and production plans of all other rms (imitators, Cournot players but rational players too). The rational rms produce epectations over all possible mitures of heuristics resulting from randomly drawing the n other players from the large population of rms. In this setting the rational objective function, its own epected utility is of the following form: U R t (q i;t j) = X f k ;k 2 (n ; R ; C ) P (k q R t + k 2 q C t + (n k k 2 )q I t + q i;t )q i;t C(q i;t ); (42) 30

32 with f k ;k 2 (n ; R ; C ) = (n )! k!k 2!(n k k 2 )! R k t t C k 2 ( t R t C ) n k k 2 and = qt R ; qt I ; qt C ; t R ; t C. The rst order condition for an optimum of (42) is characterized by equality between marginal cost an epected marginal revenue. Given the value of q C t q I t R t C t, all rational rms coordinate on the same output level q R t. Di erentiating equation (42) with respect to q i;t gives the rst order condition, which is equal for all rational rms. This rst order condition is given by: U R t (q i;t j) q i;t = 0 which equals to: X f k ;k 2 (n ; R ; C ) [P ((k + )qt R + k 2 qt C + (n k k 2 )qt I )+ P 0 ((k + )qt R + k 2 qt C + (n k k 2 )qt I )qt R C 0 (qt R )] = 0 (43) Let the solution to this be given by q R t = H R (q C t ; q I t ; R t ; C t ). The system of quantity dynamics is thus given by q R t+ = H R (q C t+; q I t+; R t+; C t+) q C t+ = R((n )( R t q R t + C t q C t + ( R t C t )q I t ) (44) q I t+ = R t q R t + C t q C t + ( R t C t )q I t Note that rational player plays such that epected marginal revenue equals marginal cost at t + and a Cournot rm plays such that its marginal revenue (of period t) equals marginal cost (at period t). Therefore the Cournot heuristic is a lagged version of the rational heuristic 3

33 if and only if P 0R q R + C q C + ( R C )q I ) + q C ) = X f k ;k 2 (n ; R ; C )P 0 ((k + )q R + k 2 q C + (n k k 2 )q I ): (45) Thus the Cournot heuristic is only a lagged version of the rational heuristic if the inverse demand is linear. In this speci c case the analysis become easier because this gives the possibility to lower the dimension of the dynamical system. It is easily checked that if the imitation and Cournot rms play the Cournot-Nash equilibrium quantity q, or if all rms are rational, the rational rms will play the Cournot-Nash equilibrium quantity, that is H R (q ; q ; t R ; t C ) = q, for all R and C and H R (qt+; C qt+; I ; 0) = q for all q C, q I. In the remainder we will denote by H R q, H R R q, H R C q, H R I and H R R the partial C derivatives of H R (q C ; q I ; R ; C ) with respect to q R, q C, q I, R and C respectively, evaluated at the equilibrium (q ; q ; q ; R C ), which we will denote by in the remainder of this chapter for notational convenience. Now that we have the quantity dynamics we can turn to the population dynamics. These are related to the population dynamics from Section 4 but di er signi cantly since we are in a three heuristic environment now. The population dynamics, as in Section 4, depend on relative tness. Let the fraction dynamics be given by R;t+ = K R (U R t ; U C t ) C;t+ = K C (U R t ; U C t ): (46) Where t+is R the fraction of rational rms in period t+ whereas t+ C is the fraction of Cournot rms in that period. With Ut R = R t C R ( C t C C ) we denote the di erence in average tness of the rational and the Cournot heuristic and with Ut C = C t C C ( I t C I ) we denote the di erence in average tness of the Cournot and the imitation heuristic. Note that K R and K C are R 2! [0; ] are continuously di erentiable functions where the di erence in 32

34 tness of the rational and Cournot heuristics and the di erence in tness of the Cournot and imitation heuristic are used as input. The di erence in tness of the rational and imitation heuristic is not used as an input variable since this information is captured implicitly in the other two di erences. Note that K R is a monotonically increasing function in the rst and second element whereas K C is decreasing in the rst element but increasing in the second element. Furthermore, K R (0; 0) = K C (0; 0) =. In the remainder of this chapter we denote 3 K R and K R 2 the partial derivatives of K R with respect to the rst and the second element respectively and with K C and K C 2 the partial derivatives of K C with respect to the rst and the second element respectively. Now that we have the quantity and population dynamics, we know the full system of equations. The full system is given by: q R t+ = = H R ( 2 ; 3 ; 4 ; 5 ) q C t+ = 2 = R((n )( R t H R (q C t ; q I t ; R t ; C t ) + C t q C t + ( R t C t )q I t ) q I t+ = 3 = R t q R t + C t q C t + ( R t C t )q I t (47) R t+ = 4 = K R (U R t ; U C t ) C t+ = 5 = K C (U R t ; U C t ): Since a dynamical system can only depend on lagged variables we substituted 2 ; 3 ; 4 ; 5 into H R (). In order to determine the local stability of the unique equilibrium, we need to determine the eigenvalues of the Jacobian matri evaluated at that equilibrium. It can easily be shown that the partial derivatives of 3 with respect to q R, q C, q I, R and C, evaluated at the equilibrium are R, C, R C, 0 and 0 respectively. To determine the partial derivatives of 4 and 5 we need to determine the partial derivatives of U R t and U C t. In accordance to Section 4.2 we can write the rst pro t di erential as U R = X A k ( R ; C )D k (q R t ; q C t ; q I t ; R ; C ) C R + C C ; 33

35 with A k ;k 2 ( R t ; C t ) = upon the produced quantities, and (n )! k!k 2!(n k k 2 )! R k t t C k 2 ( t R t C ) n k k 2, which does not depend D k ;k 2 (q R t ; q C t ; q I t ; R t ; C t ) = P ((k + )q R t + k 2 q C t + (n k k 2 )q I t )q R t C(q R t ) [P (k q R t + (k 2 + )q C t + (n k k 2 )q I t )q C t C(q C t )]: (48) Which depends upon R and C through q R t = H R (q C t ; q I t ; R t ; C t ). Note that D k ;k 2 (q ; q ; q ; R t ; C t ) = 0; 8 R ; C. Net to that the partial derivatives of D k ;k 2 () evaluated at the equilibrium are given by D k (qt R ; qt C ; qt I ; R ; C ) qt R = [P 0 )q + P (Q ) C 0 )]H R q ( ) = 0 R D k (qt R ; qt C ; qt I ; R ; C ) qt C = [P 0 )q + P (Q ) C 0 )](H R q ( ) ) = 0 C D k (qt R ; qt C ; qt I ; R ; C ) qt I = [P 0 )q + P (Q ) C 0 )]H R q ( ) = 0 I D k (qt R ; qt C ; qt I ; R ; C ) R = [P 0 )q + P (Q ) C 0 )]H R( ) = 0 D k (qt R ; qt C ; qt I ; R ; C ) C = [P 0 )q + P (Q ) C 0 )]H C( ) = 0: (49) Where the second equalities follow from the fact that P 0 )q + P (Q ) C 0 ) = 0 is the rst order condition of any rm in a Cournot-Nash equilibrium. Using this it follows immediately that the partial derivatives of 4 are given by: 4 q R t = K R (C C C R ; C I C C ) U R = 0 q R t + K2 R (C C C R ; C I C C ) U C q R t (50) and 4 q C t = K R (C C C R ; C I C C ) U R = 0 q C t + K2 R (C C C R ; C I C C ) U C q C t (5) 34

36 and 4 q I t = K R (C C C R ; C I C C ) U R = 0 q I t + K2 R (C C C R ; C I C C ) U C q I t (52) and 4 R t = K R (C C C R ; C I C C ) U R = 0 R t + K2 R (C C C R ; C I C C ) U C R t (53) and 4 C t = K R (C C C R ; C I C C ) U R = 0: C t + K2 R (C C C R ; C I C C ) U C C t (54) Furthermore, the partial derivatives of 5 are given by 5 q R t = K C (C C C R ; C I C C ) U R = 0 q R t + K2 C (C C C R ; C I C C ) U C q R t (55) and 5 q C t = K C (C C C R ; C I C C ) U R = 0 q C t + K2 C (C C C R ; C I C C ) U C q C t (56) and 5 q I t = K C (C C C R ; C I C C ) U R = 0 q I t + K2 C (C C C R ; C I C C ) U C q I t (57) 35

37 and 5 R t = K C (C C C R ; C I C C ) U R = 0 R t + K2 C (C C C R ; C I C C ) U C R t (58) and 5 C t = K C (C C C R ; C I C C ) U R = 0: C t + K2 C (C C C R ; C I C C ) U C C t (59) The Jacobian of the system, evaluated at the equilibrium is thus given by 0 H R q H R R q H R C q H R I H R R C J 2 J 22 J 23 J 24 J 25 Jj = R C R C 0 0 B A (60) with J 2 = (n ) R H R q R R 0 ) J 22 = (n ) R H R q C + C R 0 ) J 23 = (n ) R H R q I + R C R 0 ) J 24 = (n ) H R + R H R R q R 0 ((n )q ) J 25 = (n ) H R + R H R C q R 0 ((n )q ) : This Jacobian has very complicated eigenvalues which cannot be epressed in a useful function, for this we have to look at the leading eample. Leading eample. We know that the Cournot heuristic is a lagged version of the rational 36

38 heuristic in this leading eample since the inverse demand function is linear, therefore the dimension of the dynamical system can be reduced by one. Note that only the Cournot production is a lagged version of the rational production. The Cournot pro ts and resulting fractions are in general not lagged rational pro ts and fractions. The production plans of the system, are given by qt+ R = H R a c n ( t ) = ( b(2 + (n )t+) R 2 + (n ) t+q C C t+ R t+ + ( t+ R t+)q C t+) I qt+ C = a c 2b 2 (n )(R t qt R + t C qt C + ( R C )qt I ) q I t+ = R t q R t + C t q C t + ( R C )q I t ; (6) where t = (q C t ; q I t ; R t ; C t ). Furthermore, the average pro t (Eq. (5)) boils in the leading eample down to R t =F (H R ( t ); q C t ; q I t ; R t ; C t ) =(a c)h R ( t ) b(h R ( t ) + (n )q I t )H R ( t ) (62) b H R ( t ) q I t (n ) R t H R ( t ) b(q C t q I t )(n ) C t H R ( t ); using that X X X (n )! k!k 2!(n k k 2 )! R k C k 2( t t R t t C ) n k k 2 = ; (n )! k!k 2!(n k k 2 )! R t k C t k 2( R t C t ) n k k 2 k = (n ) R t ; (n )! k!k 2!(n k k 2 )! R k C k 2( t t R t t C ) n k k 2 k 2 = (n )t C : (63) Remember that C t = F (q C t ; H R ( t ); q I t ; C t ; R t ) and I t = F (q I t ; q C t ; H R ( t ); R t C t ; C t ). For the population dynamics we use the Logit dynamics, as for eample discussed in Brock and Hommes (997). The complete dynamical system in this leading eample is thus given 37

39 by 4 q C t+ = ( t ) = a c 2 + R (n ) 2b 2 (n )(C t qt C + ( R C )qt I ) q I t+ = 2 ( t ) = R t () + C t q C t + ( R C )q I t R t+ = 3 ( t ) = C t+ = 4 ( t ) = e U R t e U R t + + e U C t e U C t e (U R t +U C t ) + e U C t + : (64) This system has one unique equilibrium where all rms produces the Cournot-Nash quantity q. Since production is equal at the equilibrium, pro ts are equal at the equilibrium. The equilibrium fractions are therefore a function of the information costs and the evolutionary pressure, given by R = e (CC C R ) e (CC C R ) + + e (CI C C ) and C = e (CI C C ) e (CI C R ) + e (CI C C ) + : The Jacobian of the system in the leading eample evaluated at the equilibrium is therefore given by 0 C Jj = (n ) C 2+ R (n ) (n )( R C ) J 2+ R (n ) 3 J 4 ( R C (n ) ) R R (n ) ; (65) C A (n ) R 2+ R (n ) For the calculation of the eigenvalues J 3 and J 4 are irrelevant because the third and fourth 4 The population dynamics can alternatively be epressed as e U R;t t+ R = e U R;t + e U C;t + e U I;t t+ C e U C;t = e U R;t + e U C;t + e U I;t ; which is a more common but equivalent epression. The rational production plan is left out of the dynamical system because the Cournot production plan is a lagged version of the rational production plan. 38

40 row contain only zeros. For general R and C the eigenvalues become very lengthy epressions which cannot be simpli ed. Nevertheless, because the rational heuristic uses much more information than the Cournot and the imitation heuristic, we set the cost of this heuristic equal to C R > 0, while we set the cost of the other heuristics equal to 0, without loss of generality. After this parameterization we can calculate the eigenvalues analytically. The eigenvalues are a function of n and the the product of and C R. The eigenvalues are given by = 2 = 3 = 0 and 4 (n; C R ) = 3eCR ne CR n + 4e CR + For the system to be stable we need j 4 (n; C R )j <. Note that 4 is always less than. Rearranging gives that the threshold number of rms is given by n < (C R ) = 7eCR + e CR (66) Note that economically the number of rms can only be an integer but mathematically the number of rms can be treated as a continuous variable. From equation (66) we see that when information cost C R is close to zero, the system is stable for all n. When C R = and = 3, as simulated below, the equilibrium a c c is stable when n < 7:42. When n = 7:42, the system undergoes ; a ; e 3 ; b(n+) b(n+) e 3 +2 e 3 +2 its rst bifurcation. The largest eigenvalue is equal to at the bifurcation, indicating that the rst bifurcation is a period-doubling bifurcation. This is con rmed by the simulations below. 39

41 (a) Bifurcation diagram (q t ; n) (b) Time path of Cournot and imitiation quantities (c) Cournot pro t di erential (d) Time path Cournot fraction (e) Largest Lyapunov eponent (f) Largest Lyapunov eponent Figure 4: Linear n-player Cournot competition between rational, Cournot and imitation rms with endogenous fraction dynamics. The leading eample is simulated in the Fig. 4. Panel (a) depicts the bifurcation diagram for increasing number of rms n. The rst period-doubling bifurcation appears, as calculated analytically for n = 7:42. For n = :85, the system undergoes a Hopf-bifurcation which creates highly non-linear dynamics. For 3 n 4:4, the system is in a 0-cycle whereas for n > 4:4 the system becomes chaotic again. Panel (b) displays oscillating time series of produced quantity by the Cournot and imitation rms and the equilibrium quantity fraction 40